such that 
.
such that 
and 
.
.
such that the inequality below holds for any positive real numbers 
:![\[ \frac ab + \frac bc + \frac ca - 3 \geq k \left( \frac a{b + c} + \frac b{c + a} + \frac c{a + b} - \frac 32 \right).
\]](http://latex.artofproblemsolving.com/2/5/c/25c95e3e933649ed55bdd49ee446475223d8c871.png)
![$f \in R[X]$](http://latex.artofproblemsolving.com/b/5/6/b56c526a97b7f343d66e1317d98611980543ff21.png)
has an odd degree and it is given that 
is co-prime with 
and
Prove that 
there exists a positive integer N such that 
for every positive integer 
,
we denote the sum of digits of a positive integer 
.
be a sequence of rational numbers given by 
and for all 
we have 
.
can be partitioned into 
parallelograms. Prove that it can also be partitioned into 
parallelograms.
Y by
Let 
be two positive integers, 
, for all non-negative integers 
and 
random decimal
digits with 
. Prove that there exists some positive integer 
for which the representation in the decimal base of the number 
begins with the digits in this order.
lesser or equal with 
be two positive integers, 
, for all non-negative integers 
and 
random decimal
digits with 
. Prove that there exists some positive integer 
for which the representation in the decimal base of the number 
begins with the digits in this order. lesser or equal with 
